Question

The value of $\int e^{sinx}sin 2xdx$ is

• A. $2e^{sinx} (sin x - 1) +C$
• B. $2e^{sinx} (sin x + 1) +C$
• C. $2e^{sinx} (cos x + 1) +C$
• D. $2e^{sin x} (cos x - 1) +C$

Answer 1

Answer: (A)

$\left(I = \int e^{sin\,x} \ sin \,2x \ dx = 2 \int e^{sin\,x} \ sin \,x \ cos \,x \ dx\right)$
Let $t=\sin\, x$
$\Rightarrow d t=\cos \,x d x$
Therefore, $I=2 \int {te}^{t} d t$
$=2\left(t e^{t}-e^{t}\right)+C$
$=2(\sin \,x-1) e^{\sin\, x}+C$